Prandtl-Batchelor flows with a point vortex on a disk

In this paper we aim to construct a very weak solution to the steady two-dimensional Navier-Stokes equations which is affected by an external force induced by a point vortex on the unit disk. Such a solution is also the form of Prandtl-Batchelor type, i.e. the vorticity in the limit of vanishing viscosity is constant in an inner region separated from the boundary layer. Multi-scale asymptotic analysis which will capture well the singular behavior of the solution is used to construct higher order approximate solutions of the Navier-Stokes equations firstly, and then stability analysis is performed for the error system, finally the existence of Prandtl-Batchelor flows with a point vortex on a disk with the wall velocity slightly different from the rigid-rotation is proved. To overcome the singularity at the origin from the point vortex and strong singular behaviors near the boundary from viscous flow, we introduce a coordinate transformation which converts a bounded singularity problem to an unbounded one, and use a new decomposition for the error and perform basic energy estimate and positivity estimate for each part of the decomposition separately. The equation in the vorticity form shall be used to deal with $L^\infty$ estimate of the vorticity error.